Theorem eqer 6138

Description: Equivalence relation involving equality of dependent classes A ( x ) and B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
eqer.1	|- ( x = y -> A = B )
eqer.2	|- R = { <. x , y >. | A = B }

Assertion

Ref	Expression
eqer	|- R Er _V

Distinct variable groups:   x, y   y, A   x, B

Allowed substitution hints:   A( x)   B( y)   R( x, y)

Proof of Theorem eqer

Dummy variables w z v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.2	. . . . 5 |- R = { <. x , y >. | A = B }
2	1	relopabi 4463	. . . 4 |- Rel R
3	2	a1i 9	. . 3 |- ( T. -> Rel R )
4		id 19	. . . . . 6 |- ( [_ z / x ]_ A = [_ w / x ]_ A -> [_ z / x ]_ A = [_ w / x ]_ A )
5	4	eqcomd 2045	. . . . 5 |- ( [_ z / x ]_ A = [_ w / x ]_ A -> [_ w / x ]_ A = [_ z / x ]_ A )
6		eqer.1	. . . . . 6 |- ( x = y -> A = B )
7	6, 1	eqerlem 6137	. . . . 5 |- ( z R w <-> [_ z / x ]_ A = [_ w / x ]_ A )
8	6, 1	eqerlem 6137	. . . . 5 |- ( w R z <-> [_ w / x ]_ A = [_ z / x ]_ A )
9	5, 7, 8	3imtr4i 190	. . . 4 |- ( z R w -> w R z )
10	9	adantl 262	. . 3 |- ( ( T. /\ z R w ) -> w R z )
11		eqtr 2057	. . . . 5 |- ( ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) -> [_ z / x ]_ A = [_ v / x ]_ A )
12	6, 1	eqerlem 6137	. . . . . 6 |- ( w R v <-> [_ w / x ]_ A = [_ v / x ]_ A )
13	7, 12	anbi12i 433	. . . . 5 |- ( ( z R w /\ w R v ) <-> ( [_ z / x ]_ A = [_ w / x ]_ A /\ [_ w / x ]_ A = [_ v / x ]_ A ) )
14	6, 1	eqerlem 6137	. . . . 5 |- ( z R v <-> [_ z / x ]_ A = [_ v / x ]_ A )
15	11, 13, 14	3imtr4i 190	. . . 4 |- ( ( z R w /\ w R v ) -> z R v )
16	15	adantl 262	. . 3 |- ( ( T. /\ ( z R w /\ w R v ) ) -> z R v )
17		vex 2560	. . . . 5 |- z e. _V
18		eqid 2040	. . . . . 6 |- [_ z / x ]_ A = [_ z / x ]_ A
19	6, 1	eqerlem 6137	. . . . . 6 |- ( z R z <-> [_ z / x ]_ A = [_ z / x ]_ A )
20	18, 19	mpbir 134	. . . . 5 |- z R z
21	17, 20	2th 163	. . . 4 |- ( z e. _V <-> z R z )
22	21	a1i 9	. . 3 |- ( T. -> ( z e. _V <-> z R z ) )
23	3, 10, 16, 22	iserd 6132	. 2 |- ( T. -> R Er _V )
24	23	trud 1252	1 |- R Er _V

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   T. wtru 1244   e. wcel 1393   _Vcvv 2557   [_csb 2852   class class class wbr 3764   {copab 3817   Rel wrel 4350   Er wer 6103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-er 6106

This theorem is referenced by:  ider  6139